Math 381 - Discrete Mathematical Modeling

Lecture Summaries

10W - 03/07/12

• Various distributions, continuous distributions.
• The Poisson distribution and the Exponential distribution.
• Queueing Theory and the Exponential distribution.
• Generating customer arrivals using the Poisson distribution.
• R code for the Fishtank example using a Poisson distribution.
• Analysis of the results of a run of simulations.
• R code to repeat the Fishtank example and analyze the results.
• Confidence intervals.

10M - 03/05/12

• The Fishtank example: simulating inventory problems.
• Structure of a simulation, strategies.
• Expected value, confidence intervals.
• R code for the Fishtank example.
• The Bernoulli distribution, the Binomial distribution.
• The Poisson distribution.

9F - 03/02/12

• More comments on the homework.
• The Stock Market example: analyzing data in R.
• Sampling from a discrete distribution.
• Modeling trading stategies.

9W - 02/29/12

• Comments on the homework.
• The sample() function.
• Simulating a card game: code for the Poker example.
• Using the sum() function on a boolean list.
• Creating a discrete distribution from historical data: the hist() function.
• The Stock Market example: code.

9M - 02/27/12

• Simulations: Sampling from a discrete distribution.
• The runif() function and pseudo-random numbers.
• The Dice example: here's the code.
• The Dice Game example: code.
• Loops in R, the switch statement.
• Sampling from a non-uniform distribution.
• The Law Firm example: code.
• Using R to analyze data.
• Read Chapter 4 in the old course notes.

8F - 02/24/12

• More comments on the projects.
• Basic probability: Discrete random variables.
• Sample space and Distribution.
• Events: Independent or Mutually exclusive
• Statistics: Expected value and Standard deviation.
• Simulations.
• Read these notes on probability. Also read Chapter 3 in the old course notes.
• Homework due Wednesday, 2/29: Old course notes Ch. 3 #1,2,4,6,7

8W - 02/22/12

• Comments on the projects.
• The Worker scheduling problem: more constraints.
• Maximum number of block per day.
• Must work two hours in a row.
• Some logic.
• Questions about feasibility.

7F - 02/17/12

• Comments on the homework.
• Please consider volunteering for Math Day.
• The Worker scheduling problem: here's the paper.
• The Objective function.
• Max hours in a day.
• All hours in a day or none.
• No large spread.

7W - 02/15/12

• Comments on the projects.
• Email me by next Wednesday if you want to choose your partner for the 3rd one.
• Either/or constraints: the Auto Company example.
• Worker scheduling and the transport matrix.

7M - 02/13/12

• Branch and bound techniques for solving the TSP.
• Use the greedy algorithm to get an upper bound. (Any Ham cycle will do.)
• The tree splits the decision space. Fathom all branches to get a lower bound.
• Exotic constraints in Integer Programming: Either/or constraints.
• Read section 9-2.

6F - 02/10/12

• Subtour elimination constraints: here's a good paper.
• Second projects due next Friday.
• Strategies for solving a TSP.
• CPLEX form of an IP: here's the input file for the 5 cities.
• GUSEK works well on a PC.
• Using GLPK from the command line.
• Interpreting the GLPK output file.

6W - 02/08/12

• Subtour elimination constraints.
• You can read about these in section 9-6 in Winston.
• lpSolve works well to solve IPs in R.
• You can also use GLPK to solve larger IPs in R.
• I can easily solve very large IPs with GUSEK.
• The NEOS Server can solve your big IP.
• More homework due 02/13/2012: 9-6 #10

6M - 02/06/12

• Minimum Spanning Trees: Prim's algorithm.
• A lower bound for the TSP.
• The Euclidean TSP and MST's: an upper bound.
• The Transport matrix and total unimodularity.
• More on structured matrices in R: the 5 cities.
• A greedy algorithm gives an upper bound to the TSP.
• Here's the code I was talking about today.
• Attend the MAC talk on mathematical biology on Friday (2:30 in Kane 210).
• Read 9-1 and the part of 9-6 on the TSP.
• Also read section 2.5 in the Old Course Notes
• Homework due 02/13/2012: 9-6 #4 and Problem #7 in section 2.8 of the Old Course Notes.

5F - 02/03/12

• Comments on the papers: grading criteria, grammar notes.
• Topics for the next project: the Traveling Salesman Problem.
• The TSP: a 5 city example.
• Some remarks on Graph Theory: Hamiltonian cycles.
• A greedy algorithm gives an upper bound to the TSP.
• Modeling the TSP as an Integer Program: the Transport Matrix.
• Using R to create the Transport Matrix.
• The Georgia Tech TSP webpage: Take a look at the games.

5W - 02/01/12

• Comments on the papers: references, descriptions of the model.
• Topics for the next project: Markov chain model of a game.
• Absorbing Markov chains: the Dice game.
• Ideas behind the proofs of facts about absorbing Markov chains.
• Example with interesting states: the Coin game.

5M - 01/30/12

• New groups will go up today.
• Comments on the papers: descriptions of the model, results, length and conciseness.
• Absorbing Markov chains: the Dice game.
• The Fundamental matrix: expected length of the game.
• Calculating the probability of winning.
• Raising a matrix to a power in R.
• Here's my R code for all this.
• Read 17-6.
• More homework due 2/3/12: 17-6 #1,2

4F - 01/27/12

• Markov chains: The transition matrix acts on the right.
• Projects: the Diet problem.
• Infeasible linear programs: strategies in R.
• Here's the code I ran in class.
• Absorbing Markov chains: the Dice game.

4W - 01/25/12

• The read.csv function is a good way to get data into R.
• Ergodic Markov chains: aperiodicity.
• The Cola example and steady state probabilities.
• Mean first passage times.
• Read 17-1 through 17-5.
• Homework due 2/3/12: 17-2 #3; 17-3 #1; 17-5 #4,7

4M - 01/23/12

• Comments on the format of your paper.
• I gave an A+ to this paper the last time I taught the class.
• Here is another good paper.
• Both of these examples are much longer than your paper will be.
• Markov Chains.
• Ergodic Markov chains: the Cola example.
• The steady state of an ergodic Markov chain.

2F - 01/13/12

• Comments on the homework
• Dijkstra's Algorithm in R.
• Basic graph stuff: plot a graph.
• Dijkstra and adjacency matrices: sample code.
• Dijkstra and edge lists: sample code.
• This is a helpful URL for getting graph theory working in R.
• Information on graph algorithms in R: the RBGL package.
• Read sections 8-1 and 8-2 in the text.
• Homework due Friday: 8-2 #1,2,5

• Choose a project and start gathering data.
• Consider volunteering for Math Day. Email Jim Morrow if you are interested.
• Graph Theory: Basic Definitions.
• The Shortest Path Problem and Dijkstra's Algorithm.
• Read section 8-1 and 8-2 in the text. Skip the section on the "Transshipment Problem".
• The Wikipedia article on Graph Theory is quite good.
• The LyX typesetting program is a nice LaTeX package.

2M - 01/09/12

• The textbook will be in the Bookstore on Thursday.
• Here are the chapters we are covering this week: Chapter 3 and Chapter 8.
• I've posted the groups for the 1st project here.
• Matrix form of a linear program.
• More linear programming: The diet problem. Here's some R code to solve this problem: diet.R
• Solving Integer Programs with R. See the Lifeboats example from 01/06/12.
• Homework due Friday. These problems are in the textbook 3-2 #1,5; 3-4 #2,3.
• Consider attending this interesting talk on Friday.

1F - 01/06/12

• The Bookstore says the textbook will be in next Wednesday.
• Meanwhile, here are some examples. (Click on the problem number for solutions.)
• More on linear and integer programming.
• The lifeboat example.
• Here's the R code I ran today: life-boats-2.R.
• Using R to solve LPs and IPs. You can download R for free here.
• This is a nice quick R reference.

1W - 01/04/12

• Overview of course
• Introduction to the modeling process.
• Linear Programming
• The statistical programming language R can be used to solve Linear and Integer Programs.
• You can download it for free here.
• Here is a comprehensive introduction to R.
• Here's the Syllabus
• Homework due Monday. These problems are in the textbook. Section 3-1 #1,2,4
• I've posted the problems here.